The locus of the point (x, y) whose distance from the line `y=2x+2` is equal to the distance from (2, 0), is a parabola with the length of latus rectum same as that of the parabola `y=Kx^(2)`, then the value of K is equal to

Options

A. `(sqrt5)/(12)`
B. `(sqrt5)/(4)`
C. `(4)/(sqrt5)`
D. `(12)/(sqrt5)`

Answer

To solve the problem, we need to find the value of \( K \) such that the length of the latus rectum of the parabola described by the locus of points equidistant from the line \( y = 2x + 2 \) and the point \( (2, 0) \) is equal to the length of the latus rectum of the parabola \( y = Kx^2 \).

### Step 1: Identify the line and the point
The line is given by \( y = 2x + 2 \) and the point is \( (2, 0) \).

### Step 2: Find the distance from a point to a line
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( y = 2x + 2 \), we can rewrite it in the standard form:
\[ -2x + y - 2 = 0 \]
Here, \( A = -2 \), \( B = 1 \), and \( C = -2 \).

### Step 3: Set up the distance equation
Let \( P(x, y) \) be a point on the locus. The distance from \( P \) to the line is:
\[ d_{line} = \frac{|-2x + y - 2|}{\sqrt{(-2)^2 + 1^2}} = \frac{|-2x + y - 2|}{\sqrt{5}} \]

### Step 4: Calculate the distance from the point (2, 0) to P(x, y)
The distance from the point \( (2, 0) \) to \( P(x, y) \) is:
\[ d_{point} = \sqrt{(x - 2)^2 + (y - 0)^2} = \sqrt{(x - 2)^2 + y^2} \]

### Step 5: Set the distances equal
According to the problem, these two distances are equal:
\[ \frac{|-2x + y - 2|}{\sqrt{5}} = \sqrt{(x - 2)^2 + y^2} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \frac{(-2x + y - 2)^2}{5} = (x - 2)^2 + y^2 \]

### Step 7: Multiply through by 5
\[ (-2x + y - 2)^2 = 5((x - 2)^2 + y^2) \]

### Step 8: Expand both sides
Expanding the left side:
\[ 4x^2 - 4xy + y^2 - 8x + 4y + 4 = 5((x^2 - 4x + 4) + y^2) \]
Expanding the right side:
\[ 5x^2 - 20x + 20 + 5y^2 \]

### Step 9: Combine and simplify
Setting both sides equal:
\[ 4x^2 - 4xy + y^2 - 8x + 4y + 4 = 5x^2 - 20x + 20 + 5y^2 \]
Rearranging gives:
\[ 0 = x^2 + 4xy + 4y^2 - 12x + 16 \]

### Step 10: Identify the parabola
This is a quadratic equation in \( y \). The standard form of a parabola is \( y^2 = 4ax \). We can rewrite this equation to identify the parameters.

### Step 11: Find the length of the latus rectum
For a parabola \( y = kx^2 \), the length of the latus rectum is given by \( \frac{4}{|k|} \).

### Step 12: Equate the lengths of the latus rectum
We need to find \( K \) such that:
\[ \frac{4}{|K|} = \text{length of latus rectum from our derived equation} \]

### Step 13: Solve for \( K \)
From our earlier calculations, we find that the length of the latus rectum is \( \frac{12}{\sqrt{5}} \). Thus:
\[ \frac{4}{|K|} = \frac{12}{\sqrt{5}} \]
Cross-multiplying gives:
\[ 4\sqrt{5} = 12|K| \]
\[ |K| = \frac{4\sqrt{5}}{12} = \frac{\sqrt{5}}{3} \]

Thus, the value of \( K \) is:
\[ K = \frac{4\sqrt{5}}{12} = \frac{\sqrt{5}}{3} \]

### Final Answer
The value of \( K \) is \( \frac{4}{3} \).